18. After being repaired, a machine functions for an exponential time with rate and then fails....

18. After being repaired, a machine functions for an exponential time with rate λ and then fails. Upon failure, a repair process begins. The repair process proceeds sequentially through k distinct phases. First a phase 1 repair must be performed, then a phase 2, and so on. The times to complete these phases are independent, with phase i taking an exponential time with rate μi , i = 1, . . . , k.

(a) What proportion of time is the machine undergoing a phase i repair?

(b) What proportion of time is the machine working?